Depending on how long your world needs to maintain its breathable atmosphere in your story, you might need to change the parameters of the planet to enable it to retain the atmosphere long enough. I discuss this in Part One.
And I discuss how an icey world with not photosynthetic life could form an oxygenrich atmospere in Part Two.
Part One. Escape Velocity.
As I wrote in a comment to another question today, the ability of a planet to retain for geological eras of time whatever atmosphere it acquires depends on its escape velocity, not on its surface gravity.
ESCAPE VELOCTIY NOT SURFACE GRAVITY.
As it happen, the formula to calculate the surface gravity of a planet is different from the formula to calculate the escape velocity of a planet. So changing the surface gravity of a planet by X amount will not necessarily change the escape velocity of the planet by X amount. The amount of change in the escape velocity can be more or less than the amount of change in the surface gravity.
Never use the surface gravity of a world as an indication of how well it can retain an atmosphere. Use the escape velocity instead.
So the question says the rogue planet:
has a gravity comparable to Mars' (3.7m/s²)
According to Wikipedia, Mars has a surface gravity of 3.72076 meters per second per second and an escape velocity of 5.027 meters per second.
There are a number of factors which influence the ability of a palnet to retain an atmosphere for long eras of time, and other factors which influence what atmosphere the planet acquires in the first place. But the most important factors that influences how long long a world can retain its atmosphere are its escape velocity and the ratio between that escape velocity and the velocity of atmopsheric molecules and atoms in the exosphere of the world.
The importance of that ratio in determinining how fast or slowly a world loses atmosphere is discussed by Stephen H. Dole, in Habitable Planets for Man, 1964, pages 34 & 35. The ratio of the escape velocity of a world divided by the root-mean-square of the velocity of atoms & molecules of a specific gas in the exosphere of that world determines how fast the atmosphere will be reduced to 1/e (or 0.368) of its original amount.
Table 5 on page 35 show that if the ratio is 1 or 2, the atmosphere will reduce to 0.368 of it is original amount instantly. If the ratio is 3, the atmosphere will retuce to 0.368 in a few weeks. if the ratio is 4, the atmosphere will reduce to 0.368 in several thousnd years. If the ratio is 5, the atmosphere will reduce to 0.368 in about a hundred million years. If the ratio is 6, the atmosphere will take almost infinite time to reduce to 0.368 the original amount.
So a comparatively minor change in the ratio between the escape velocity of a world and the root-mean-square of the velocity of a gas in the exosphere will determine how long it can keep its original atmosphere. Other factors can hurry up the atmospheric loss, but nothing can delay it. A world can maintain its original atmospheric density by replacing lost gases as fast as they are lost, but it will have only a finite supply of those gases for replacement ing lost particles.
The velocity of gas particles depends on their temperatures.
The average surface temperature of Earth is about 13.9 degrees Celsius.
Which is about 287.05 degrees Kelvin (K)
But the temperature of atmospheric particles in the exosphere of Earth, and thus their root-mean-square velocity, is much higher than that.
According to Dole, on page 54, the Earth's exosphere temperatures are about 1000 K to 2000K.
However, if we take as a rough approximation that maximum exosphere temperatures as low as 1000 K are not incompatible with the rquired surface conditions of a habitable planet, the escape velocity of the smallest planet capable of retaining atomic oxygen may be as low as 6.25 kilometers per second (5 X 1.25). Going back to figure 9, this may be seen to correspond to a planet having a mass of 0.195 Earth mass, a radius of 0.63 Earth radius, and a surface gravity of 0.49 g.
So if you world has maximum exosphere temperatures no more than 1000 K, and an escape velocity of 6.25 kilometers per second, five times the root-mean-square velocity of atmoic oxygen at 1000 K, it should be able to retain 0.368 of its atomic oxygen for about 100 million years. On the other hand, if your world had an escape velocity of 5 kilometers per second, four times the root-mean-square velocity of atomic oxygen at 1000 K, it would retain 0.368 of its atomic oxygen for only a few thousand years.
You want your world to have the surface gravity of Mars. If it also has the escape velocity of Mars, 5.027 kiometers per second, it should be able to retain 0.368 of its atmospheric oxygen for thousands of years, which might be enough for the purposes of your story. Or maybe not.
With an average surface temperature of - 10 degreees C, your world would have a surface temerature of 263.15 K, or about 0.9167392 of Earth's average temperature of 13.9 C or 287.05 K.
I don't know whether reducing the surface temperature of your world by about 10 percent to 0.9 of Earth's value would be enough to reduce the maximum exopshere temperature of your world to 1000 K instead of Earth's up to 2000 K.
You want your planet to get a lot of internal heat from tidal heating, which could be that it is far enough from any star for its exosphere to not be warmed up to more than 1000 K by radiation from the star. And depending on atmosphere composition, the internal heat emitted as infraread radiation from the surface of the planet might not be tapped by greenhouse gases in the atmosphere but escape into space without heating up the oxygen in the exosphere.
One way to keep tidal heating of your planet by its moon strong enough for geological eras of time is to make your planet actually a giant moon of a giant rogue explanet, and have other large moons orbiting that planet. The gravitational interactions between the moon that is the subject of your story and the other large moons will keep the moon's orbit from becoming too circularized, and thus keep the tidal heating stronger for longer.
And of course if your planet was wandering in interstellar space with a surface temperature of only about 5 or 10 degrees K for milions or billions of years and only recently entered the solar system and approached the Sun and got heated up to - 10 C or 263.15 K, it will have started losing its atmosphere only recently.
But you still might want to consider giving you planet a higher escape velocity to retain gases longer, while perhaps trying to keep the surface gravity as low as you can.
So going back to Mars, which seems to have been your model for your planet, it has the following attributes:
Mean radius 3,389.5 kilometers (& thus mean diameter of 6,779 kilometers.
Volume 1.63119 times 10 to the 11th power cubic kilometers (1 Mars volume or 0.151 Earth volume).
Mass 6.4171 times ten to the 23rd power kilograms (1 Mars mass or 0.107 Earth mass).
Density 3.9335 grams per cubic centimeter (1 Mars density or 0.7133 Earth Density).
Surface gravity 3.72076 meters per second per second (I Mars surface gravity or 0.3794119 Earth g.
Escape velocity 5.027 kilometers per second (1 Mars escape velocity or 0.449401 Earth escape velocity).
The density of water is about 1 gram per cubic centimeter, and thus the density of water is 0.1813565 of 5.514 grams per cubic centimeter, the average density of Earth.
Suppose that you imagine a planet with the radius, volume, density, and mass of Mars. It owuld have 0.107 the mass of Earth in 0.151 the volume of Earth. Imagine that enough water is added to Mars to double the radius of Mars, and thus increase the volume of Mars by eight times. The imaginary planet will now have 0.107 times the mass of Earth plus the volume of water in 7 times the volume of Mars, or 0.749 the Volume of Earth. So the mass of the imaginary planet will be 0.107 Earth mass plus the mass of 0.749 Earth volume in water, which should be 0.135836 the mass of Earth.
So the solid core of that planet plus the water shell around it would total 0.242836 the mass of Earth. With twice the radius of Mars, that planet would have a radius of 6779 Kilometers, a bit larger than the Earth.
According to this surface gravity calculator https://philip-p-ide.uk/doku.php/blog/articles/software/surface_gravity_calc That planet would have a surface gravity of only 0.21 g, 0.21 that of Earth.
According to this online escape velocity calculator,https://www.calctool.org/CALC/phys/astronomy/escape_velocity the planet would have an escape velocity of 5.34452 kilometers per second.
So that planet would have less surface gravity than asked for in the question, and a little higher escape velocity than Mars.
So suppose that the solid part of the planet has a radius of 3,389.5 kilometers and the water layer has a depth of 1,000 kilometers. The planet will have a total radius of 4,389.5 kilometers, 0.6889813 of Earth's radius, which will give it a total volume of 0.3270561 Earth. The water part would have 0.1760567 the volume of Earth and thus 0.031929 the mass of Earth. So the total mass of the planet would be 0.138929 Tha tof Earth. With a radius of 4,389.5 kilometers, the planet will have a surface gravity of 0.3 g and an escape velocity of 5.02371 kilometers per second.
So suppose that the solid part of the planet has a radius of 3,389.5 kilometers and the water layer has a depth of 1,510.5 kilometers. The planet will have a total radius of 4,900 kilometers, 0.76911 of Earth's radius, which will give it a total volume of 0.4549517 Earth. The water part would have 0.3039517 the volume of Earth and thus 0.0551236 the mass of Earth. So the total mass of the planet would be 0.1621236 That of Earth. With a radius of 4,900 kilometers, the planet will have a surface gravity of 0.28 g and an escape velocity of 5.13624 kilometers per second.
And there are many other variations which could be ried to get a world with a surface gravity of 3.7276 meters per second per second, or 0.3794 g, if that is desired, and an escape velocity high enough to retain an oxygen rich atmosphere for as long as is necessry for the purposes of your story.
Part Two: Getting Oxygen into the Atmosphere.
The requirements for the world in your story include:
it has a breathable atmosphere
it doesn't have photosynthetic life
On Earth the oxygen in the atmosphere was produced by photosynthetic life. Without photosynthetic life your would won't be able to produce an oxygen rich atmosphere the same way Earth did. So another way for your world to acquire an an oxygen rich atmosphere is necessary.
Fortunately another requirement is:
it is covered by water ice
As I suggested in Part One, a thicker or thiner layer of liquid water or frozen ice, or frozen ice on top of liquid water, might be necessary to adjust the overall density of your planet to give it the required surface gravity of Mars with a somewhat higher escape velocity than Mars to retain the atmosphere longer, depending on how long the world has to retain that atmosphere in your story.
In our solar system there are many worlds covered with ice of water, often mixed with ice of other compounds, and most of them are either totaly ice or else every thick layers of ice over rocky cores.
Six worlds in are solar system are believed to have global subsurface oceans under thick layers of a ice and above the rocky cores of their worlds, and another ten worlds are presently suspected of having global ocens under thick layers of ice.
If an ice covered world has no atmosphere, surface ice will gradually become water wapor, faster in warmer temperatures. If enough water vapor accumulates in the atmosphere, ice which become warm enough will not evaporate but will melt into liquid surface water.
Ultraviolet rays from the star in a star system might break up molecules of water vaport into atoms of hydrogen and oxygen. Those atoms might recombine to form new wter molecules, or they might escape into outer space. Hydrogren atoms & molecules would escape from teh world much faster than oxygen atoms & molecules wuld. Thus if various conditions were right, oxygen might accumulate in the atmosphere of ann icy world, perhaps becoming dens eenough to be brathable for humans.
If the world has an atmosphere of other gases such as nitrogen, carbon dioxide, etc., some water ice might still become vapor - it happens in earth's thick atmosphere - and be separated into oxygen and hydrogen. Earth slowly loses hydrogen that way. So that is one method for an alien, ice covered planet to convert some of its ice into hdrogen which escpes from the palnet and oxygen which might accumulate in the atmosphere until it becomes a breathable atmosphere.
And perhaps your world orbited a star with at least one companion world for many millions of years and was warmenough for an oxygen rich atmosphere to form. It might have been a planet with a large moon, or a giant moon of a giant planet.
Then a close enocunter with another planet exjected the planet (and any companions) into interstellar space. The rogue planet soon became very cold and its atmosphere liquified and then solidified.
And after millions or billions of years the world happened to enter our solar system and approach the Sun and get warmed up a lot, if the surface temperature of - 10 C or 263.15 K) is due to Sun light. And as the surface warms, the frozen atmosphere gradually liquifies and then evaporates and the planet has an atmosphere again.